$A, B$ and $C$ are sets with $A\times B=A\times C$. For which sets $A$ it follows $B=C$?

Question

For which sets $A$ does it follow that if $A\times B=A\times C$ then $B=C$?

I must prove that there are some sets $A$ that shows that $B=C$. I really don't know where to start.

Answer 1

Hint If $A \times B =A \times C$ and $a \in A$ then $$\{ a \} \times B \subset A \times B=A \times C$$

This implies that $B \subset C$. Same way you get $C \subset B$:

Details: Let $b \in B$. Then $(a,b) \in A \times B= A \times C$. Since $(a,b) \in A \times C$ we get by definition that $b \in C$.

Therefore, as long as you can pick some $a \in A$ you can prove $B=C$.